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DOI 10.1007/s00012-013-0251-2 Published online August 20, 2013

© Springer Basel 2013 Algebra Universalis

The class of algebraically closed p-semilattices is finitely

axiomatizable

Jo¨el Adler, Regula Rupp, and J¨urg Schmid

Abstract. We prove our title, and thereby establish the base for a positive solution of Albert and Burris’ problem on the finite axiomatizability of the model companion of the class of all pseudocomplemented semilattices.

1. Introduction

The purpose of this paper is to prove its title. It is based on [11] and essentially combines, in a new setting, results of Regula Rupp’s PhD Thesis [10] with results of Joel Adler’s PhD Thesis [1].

The motivation for this work comes from the problem posed by Albert and Burris in the final paragraph of [3]: “Does the class of pseudo-complemented semilattices have a finitely axiomatizable model companion?”

Together with Adler’s 2012 preprint [2], the present paper will provide a positive answer to Albert and Burris’ question. In fact, we show here that the class of all algebraically closed pseudocomplemented semilattices—for short: a.c. p-semilattices—is finitely axiomatizable in the first-order language of p-semilattices, by providing four axioms—one of which is distributivity— characterizing this class.

Recall that the model companion mentioned above consists precisely of all existentially complete—for short, e.c.—p-semilattices and is thus a subclass of the class of all a.c. p-semilattices. Adler’s preprint [2] provides finitely many additional axioms singling out the e.c. members within all a.c. p-semilattices, and thus will settle the problem.

The paper is organized as follows. Section 2 collects the basic algebraic notions concerning p-semilattices, while Section 3 provides a short summary of the relevant model-theoretic concepts, adapted to our setting.

In Section 4, we consider distributive meet-semilattices. The main result of the section is that in a distributive p-semilattice P , an arbitrary—not neces-sarily distributive—finite p-subsemilattice F  P can be extended to a finite distributive p-semilattice F0 such that F  F0 P .

Presented by J. Berman.

Received December 11. 2012; accepted in final form January 28, 2013.

2010 Mathematics Subject Classification: Primary: 03C60; Secondary: 03C05, 03C10, 03G10, 06A12.

Key words and phrases: pseudocomplemented semilattice, algebraic and existential

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In Section 5, we specify an axiom (A1) guaranteeing that F0, as obtained in Section 4, can be extended to a finite distributive p-semilattice F1 with F0F1P such that the dense elements of F1form a boolean meet-semilattice under the induced order.

In Section 6, another axiom, (A2), is introduced, and it is shown that F1, as obtained in Section 5, can be extended to a finite p-semilattice F2 with F1 F2 P such that F2 is isomorphic to a direct product of subdirectly irreducible p-semilattices, provided P satisfies (A2).

In Section 7, it is shown that in a p-semilattice P satisfying an additional axiom (A3), any finite p-subsemilattice F2, as obtained in Section 6, can be extended to a p-subsemilattice F3 P isomorphic to a direct product with finitely many factors, each of them being either the two-element boolean p-semilattice or the unique countable atom-free boolean algebra with a new top element added.

Section 8 establishes the necessity of the above axioms for a p-semilattice to be algebraically closed. Finally, Section 9 formulates our main theorem. 2. Pseudocomplemented semilattices

A pseudocomplemented semilattice (for short: p-semilattice) (P ; ∧,∗, 0, 1) is a meet-semilattice (P ; ∧) with least element 0 and top element 1, equipped with an unary operation a → a∗such that for all x ∈ P , x∧a = 0 iff x ≤ a∗. It is a nontrivial fact that the class PCS of all p-semilattices can be (finitely) ax-iomatized by identities in the first-order languageLP CS ={∧,∗, 0, 1}, making PCSa variety; see [5]. We freely write P for the p-semilattice (P ; ∧,, 0, 1) (and similarly for algebraic structures in general whenever the operations and relations under consideration are clear from the context). An element d ∈ P satisfying d∗ = 0 is called dense. D(P ) denotes the set of all dense elements of P ; moreover, (D(P ); ∧, 1) is a subsemilattice—in fact, a filter—of (P ; ∧, 1). Further, d ∈ D(P ) is called maximally dense iff d = 1 and d ≤ d ≤ 1 implies d = d or d = 1. An element s ∈ P is called skeletal iff s∗∗ = s. The set of all skeletal elements of P is denoted by Sk(P ); it is a subalgebra of the p-semilattice P . Within Sk(P ) the supremum of two elements exists w.r.t. to the order inherited from P ; in fact, supSk{a, b} = (a∗∧ b∗) for a, b ∈ Sk(P ). Setting a  b = (a∗∧ b∗), (Sk(P ); ∧, ,, 0, 1) is a boolean algebra. The set of all atoms of P is denoted by At(P ).

For any p-semilattice P , a p-semilattice ˆP is obtained from P by adding a new top element. In most cases, the top element of P will be renamed to e and 1 will stand for the new top element. We write 2 for the two-element boolean algebra and A for the unique countable atom-free boolean algebra.

The class of p-semilattices P that are generated (as p-semilattices) by their skeletal and dense elements—that is, P = Sk(P ) ∪ D(P )PCS—play an

im-portant rˆole in our context. They are called representable; equivalently, P is representable iff every x ∈ P admits a (not necessarily unique) representation

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of the the form x = x∗∗ ∧ d for some d ∈ D(P ). Obviously, in this case, P = { b ∧ d : b ∈ Sk(P ), d ∈ D(P ) }.

Although there is only one binary operation in a p-semilattice P , a notion of distributivity can be introduced: Call P distributive if for all a, b, c ∈ P with c ≥ a∧b, there exist x, y ∈ P satisfying x ≥ a, y ≥ b, and x∧y = c. Distributiv-ity in p-semilattices—in particular, its relationship with representabilDistributiv-ity—will be considered in detail in Section 4.

For a p-semilattice P and a skeletal element a ∈ P , the binary relation x θa y :⇐⇒ a ∧ x = a ∧ y is a PCS-congruence. The factor algebra P/θa is isomorphic to ({ a ∧ x : x ∈ P } ; ·,, 0, a), where (a ∧ x) · (a ∧ y) is defined as a ∧ (x ∧ y) and (a ∧ x) as a ∧ x∗. Furthermore, the map fa: P → P/θa defined by fa(x) = a ∧ x is a surjective homomorphism. The following special case will frequently occur: Consider a direct product P = i∈IPi of p-semilattices, and a subset J ⊆ I. Theni∈JPi∼=P/θa, where a ∈ P is given by (a)i = 1 iff i ∈ J, and by (a)i= 0 iff i ∈ I \ J.

In a general meet-semilattice (S; ∧), ↓Sx (or simply ↓x if S is clear from the context) stands for{ y ∈ S : y ≤ x }, the down-set generated by x in S, where x is any element of S. We write O(S) for the (distributive) lattice of all down-sets of S ordered by set inclusion. Note that if S is finite, then ↓x is actually a lattice under its induced order for any x, and we thus will call x ∈ S join-irreducible iff x is such in ↓x, for S finite. We write J (F ) for the set of all (non-zero) join-irreducibles of a finite meet-semilattice F . Whenever there is no danger of confusion,J (F ) also stands for the poset of all join-irreducibles under the order inherited from F .

Finally, a meet-semilattice (S; ∧) is called boolean iff it is the ∧-reduct of a boolean algebra. We use Q  P (respectively P  Q) freely to indicate that Q is a subalgebra of P in whatever signature P and Q are considered at the moment. More background on (p-)semilattices may be found in [5] and [7], or in [4].

3. Model theory

For a given p-semilattice P , let LPP CSbe the language obtained fromLP CS by adding bijectively a new constant symbol for each a ∈ P to LP CS. P is called algebraically closed—abbreviated by a.c.—(in PCS) if P satisfies every positive existentialLPP CS-sentence that holds in some extension P  P with P ∈ PCS. In plainer terms, P is a.c. iff every finite system of P CS-equations with coefficients from P that is solvable in some extension P  P∈ PCS al-ready has a solution in P . The stronger notion of being existentially complete— not considered in this paper but crucial in the problem posed by Albert and Burris in [3]—just differs from a.c. by allowing also (finitely many) negated equations; the model companion of PCS mentioned in the introduction is then just the class of all existentially complete algebras in PCS. For more

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background on the model theory relevant here, the reader is referred to [6], especially Chapter 7. We use ω to denote the set of all natural numbers.

In [11] the following characterization of algebraically closed p-semilattices is established.

Theorem 3.1. A p-semilattice P is algebraically closed iff for any finite sub-algebra F P , there exists a p-semilattice F isomorphic to 2r×( ˆA)sfor some r, s ∈ ω such that F  F P .

Note that the trivial one-element p-semilattice is a.c., since it only can be embedded into itself. Write A(PCS) for the class of all a.c. members of PCS. The main result of this paper is a finite list of LP CS-sentences that hold in P ∈ PCS iff P is a.c.; what actually will be shown is that these sentences hold in P iff P has the extension property specified in Theorem 3.1 above.

The remainder of this section collects some results from [1], providing evi-dence that a finite axiomatization of A(PCS) should exist. So far, the only members of A(PCS) identified immediately by Theorem 3.1 are the direct products 2r× ( ˆA)sfor some r, s ∈ ω. There are others:

Let Q be the subalgebra of ( ˆA)ωjointly generated by Sk(( ˆA)ω) and De:= { d ∈ D(( ˆA)ω) : (d)i= e for at most finitely many i ∈ ω }.

It is easy to see that Q = { a ∧ d : a ∈ Sk(( ˆA)ω) and d ∈ De}, since the latter set evidently is closed under∧ and (a ∧ d)∗∗ = a∗∗∧ d∗∗ = a∗∗∧ 1 = a∗∗ ∈ Sk(( ˆA)ω).

Note that Q is not isomorphic to any direct product with factors 2 or ˆB (B any boolean algebra), since such a product has either a finite or uncountable number of dense elements while D(Q) = Deis countable.

Let F  Q be finite. There exists a least nF ∈ ω such that (x)i = e for all x ∈ F and i > nF. Define an element a ∈ Q by (a)i = 1 for i ≤ nF and (a)i= 0 for i > nF. Let Qa= Q/θa (see Section 2) and Fa = F/θa∩ (F × F ); define Qa∗ and Fa analogously. Now Q ∼= Qa× Qa∗ canonically, Fa  Qa,

Fa∗ Qa∗, and thus F ∼=F Fa× Fa∗ for some copy F of F . It is clear that

Qa = ( ˆA)n; moreover, Fa∗ is a finite boolean subalgebra of Sk(Qa), and thus

Fa∗ = 2k for some k ∈ ω. Hence, F ∼=F Fa× Fa∗ ( ˆA)n× Fa∗ Qa× Qa.

Under the canonical isomorphism Q ∼= Qa × Qa, the algebra ( ˆA)n × Fa

corresponds to a subalgebra of Q of the form required by Theorem 3.1. That the class of all existentially complete p-semilattices—alias the model companion of PCS—can be axiomatized byLP CS-sentences follows from gen-eral model-theoretic properties of PCS, viz., the fact that PCS is a finitely generated universal Horn class with both the amalgamation and joint embed-ding properties; see [3] for details. No such general argument seems to apply to the (wider) class A(PCS). In fact, the mere axiomatizability of A(PCS) was first established in [1].

Now, an axiomatizable class of LP CS-structures is finitely axiomatizable iff both the class itself as well as its complementary class are closed under elementary equivalence and ultraproducts. So partial evidence for the finite

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axiomatizability of A(PCS) is provided by [1, Theorem 4.1], which states that an ultraproduct of finite p-semilattices that are not a.c. cannot be a.c. either.

4. Distributivity

There is a natural notion of distributivity for meet-semilattices, see Subsec-tion 4.2 below. Generally, a subsemilattice of a distributive meet-semilattice need not be distributive. However, a finite subsemilattice F  S of any dis-tributive meet-semilattice S can always be expanded to a finite disdis-tributive meet-semilattice Fsuch that F FS—a very crucial fact in our context, as we shall see. This fact is well known; to the best of our knowledge, it appeared first in print as Fact 4 in [9]. What we actually need is a p-semilattice version of this result, which does not follow immediately from [9]. Therefore, we present an exposition based on so-called minimal boolean extensions; moreover, the specific properties of such extensions will be crucial in Section 5 when they are used to construct successive distributive extensions by destroying compa-rabilities between join-irreducibles.

4.1. Minimal boolean extensions. Every semilattice (S; ∧) embeds—as a meet-semilattice—into a boolean algebra: Indeed, the map x −→ ↓x embeds S into the power set algebra P(S). If S is finite, so is P(S), and there ex-ists, therefore, a uniquely determined—up to isomorphism—smallest boolean algebra containing S as a meet-subsemilattice, denoted by BS in the sequel.

So let (F ; ∧) be an arbitrary but fixed finite meet-semilattice, and put At(BF) = {q1, . . . , qn}, thus BF = P{q1, . . . , qn}. We identify BF with its canonical copyP{q1, . . . , qn} in the sequel, and fix an embedding eF: F → BF. Given qi ∈ At(BF), define yi ∈ F by yi = { x ∈ F : qi∈ eF(x) }. The doubleton Ji:={∅, {qi}} is a nontrivial ideal in BF, so F will no longer embed into BF/Ji. With p : BF → BF/Ji the canonical (boolean) epimorphism, we thus find u = v ∈ F such that (p ◦ eF)(u) = (p ◦ eF)(v). It follows that, say, (i) qi ∈ eF(u) but (ii) qi ∈ e/ F(v). We infer (i) u ≥ yi and (ii) v  yi, so yi= u ∧ yi> u ∧ yi∧ v = yi∧ v. But

(p ◦ eF)(yi) = (p ◦ eF)(yi)∩ (p ◦ eF)(u)

= (p ◦ eF)(yi)∩ (p ◦ eF)(v) = (p ◦ eF)(yi∧ v),

and we conclude that the sets eF(yi) and eF(yi ∧ v) differ exactly in the point qi. In other words, yi has yi∧ v as its unique lower neighbor in F , that is, yi∈ J (F ).

Conversely, consider y ∈ F join-irreducible with lower neighbor y. Suppose we find atoms qi = qj in eF(y) \ eF(y). Put J = {∅, {qj}}, J is a nontrivial ideal in BF. With p : BF → BF/J the canonical epimorphism, it follows that p ◦ eF is a monomorphism, contradicting the minimality of BF. This shows that eF(y) \ eF(y) must be a singleton.

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Summing up, we have established a bijective correspondence betweenJ (F ) and At(BF), and we will identify the two sets in the sequel. This means that BF is taken to be the powerset algebra P(J (F )). Next, define OF: F → P(J (F )) by OF(x) = ↓x ∩ J (F ). Since x = sup↓xOF(x), we see that OF is injective. Also, since y ≤ x1∧ x2 iff y ≤ x1 and y ≤ x2 for x1, x2∈ F , and as y ∈ J (F ), we have OF(x1∧ x2) =OF(x1)∩ OF(x2) for all x1, x2∈ F . So, OF actually is an embedding of F into P(J (F )).

Definition 4.1. The pair (P(J (F )), OF) = (BF, OF) is the (canonical) min-imal boolean extension of a finite meet-semilattice F .

For easier reference, we also write ˆx instead of ↓x ∩ J (F ) = OF(x) for elements x of finite meet-semilattices F (there is no danger of confusion with the notation ˆP introduced for p-semilattices in Section 2).

4.2. Distributivity in meet-semilattices. The canonical notion of dis-tributivity for meet-semilattices is captured by:

Definition 4.2. A meet-semilattice S is distributive iff for all a, b, c in S, the following holds: Whenever c ≥ a ∧ b, there exist x, y ∈ S such that x ≥ a, y ≥ b, and x ∧ y = c.

It is clear that this property can be expressed by a sentence (DIST) in (the ∧-reduct of) LP CS.

The above definition of distributivity in meet-semilattices is closely related to distributivity in lattices:

Remark 4.3. For any lattice (L; ∧, ∨), its meet-semilattice reduct (L; ∧) satis-fies (DIST) iff L is distributive as a lattice. Alternatively, (S; ∧) is distributive as a meet-semilattice iff the poset of all nonempty filters of S, ordered by set inclusion, is a distributive lattice.

Note that distributivity in meet-semilattices is not necessarily inherited by subsemilattices: Let 2 be the 2-element chain 0 < 1. Then 2 × 2 \ {(1, 1)} is a nondistributive meet-subsemilattice of the distributive lattice 2× 2. Lemma 4.4. A distributive p-semilattice is representable.

Proof. Obviously, x ≥ 0 = x∗∗∧ x∗. Using distributivity we find a, b ∈ P such that a ≥ x∗∗, b ≥ x∗ and a ∧ b = x. Meeting both sides of the last equation with x∗∗ we obtain a ∧ x∗∗∧ b = x ∧ x∗∗, that is, x∗∗∧ b = x. But b ∈ D(P ), since b ≥ x, x∗ and thus b∗≤ x∗, x∗∗, that is, b ≤ 0 = x∗∧ x∗∗.  The converse of Lemma 4.4 does not hold as easy examples show. However, the distributivity of a representable p-semilattice depends only on its dense elements, as we will show presently.

For the purpose of this paper, call an element x of an arbitrary p-semilattice P distributive iff for any a, b ∈ P , x ≥ a ∧ b implies the existence of xa, xb∈ P such that xa≥ a, xb≥ b and xa∧ xb= x. It is routine to check that the meet of two distributive elements is distributive in any p-semilattice.

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Lemma 4.5. A representable p-semilattice P is distributive iff every d ∈ D(P ) is distributive.

Proof. Note first that skeletal elements are distributive in any p-semilattice: Indeed, consider a, b ∈ P and c ∈ Sk(P ) such that c ≥ a ∧ b. This implies c = c∗∗ ≥ a∗∗∧ b∗∗. By boolean distributivity, we obtain (c  a∗∗)∧ (c  b∗∗) = c  (a∗∗∧ b∗∗) = c, with c ∨ a∗∗ ≥ a∗∗ ≥ a and c ∨ b∗∗ ≥ b∗∗ ≥ b. Since P is representable, we have x = x∗∗∧ dx with suitable dx ∈ D(P ) for any x ∈ P . So x as the meet of two distributive elements is distributive provided every

d ∈ D(P ) is such. 

Given a distributive meet-semilattice S and a subsemilattice F  S, it is trivial to find a distributive semilattice F such that F  F S: Just take F = S. It turns out to be less trivial to find, for F finite, a finite distributive F extending F within S. Proposition 4.9 asserts that this is always possible. Moreover, in Proposition 4.11, we will show that the same is true within the class of all pseudocomplemented meet-semilattices.

Lemma 4.6. Let S be a distributive meet-semilattice, and let a, a1, . . . , an, b, c be elements of S.

(i) If a ∧ b ≤ c ≤ b, there exists x ∈ S such that x ≥ a and x ∧ b = c. (ii) If a1∧ c = · · · = an∧ c, there exists x ∈ S such that x ≥ ai (1≤ i ≤ n)

and x ∧ c = a1∧ c.

Proof. (i): Let a ∧ b ≤ c ≤ b. Using distributivity, we find x, y ∈ S with x ≥ a, y ≥ b, and x ∧ y = c. Since b ≥ c, we obtain c = x ∧ y = x ∧ y ∧ b = x ∧ b. (ii): Suppose a1∧ c = · · · = an∧ c and consider a1∧ c = a2∧ c. Using distributivity on a1 ≥ a2∧ c, find u2, uc ∈ S such that u2≥ a2, uc ≥ c, and u2∧ uc = a1(thus, u2≥ a1, a2). Analogously, a2≥ a1∧ c gives the existence of v1, vc ∈ S satisfying v1 ≥ a1, vc ≥ c, and v1∧ vc = a2 (thus, v1 ≥ a1, a2). Put x12= v1∧ u2; then x12≥ a1, a2. Moreover,

x12∧ c = x12∧ uc∧ vc∧ c = v1∧ u2∧ uc∧ vc∧ c = a1∧ a2∧ c = a1∧ c = ac∧ c.

Repeat this process suitably often, first proceeding with x12∧ c = a3∧ c =

· · · = an∧ c. 

Corollary 4.7. A distributive meet-semilattice S is upwards directed, that is, any two elements have a common upper bound in S. If S is also finite, then it is a distributive lattice under its natural order.

Proof. Putting c = b in Lemma 4.6(i), obtain x as a common upper bound for a and b. If S is finite, it will thus contain a greatest element, and thus the

supremum of any two elements. 

Going back to the minimal boolean extension BF of a finite meet-semilattice F , note that OF(x) is actually a down-set in J (F ). Hence, OF embeds

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F into the sublattice O(J (F )) of P(J (F )). For easier reference, we also write LF for the distributive latticeO(J (F )); LF is generated, as a lattice, by{ OF(y) : y ∈ J (F ) } and is (up to isomorphism) the uniquely determined minimal distributive lattice embedding F . By Corollary 4.7, LF can also be characterized as the (unique up to isomorphism) minimal distributive meet-semilattice embedding F . Note also that BLF =BF canonically.

Corollary 4.8. For any finite meet-semilattice F , OF provides an embedding of F into LF. Moreover, F is distributive iff OF is an isomorphism between F and LF.

4.3. Distributive extensions. The basic result, for our purposes, is: Proposition 4.9. Assume S is a distributive meet-semilattice and F  S a finite subsemilattice of S. Then there exists a finite distributive semilattice F0 such that F  F0 S. In fact, we find such F0 satisfying F0=LF.

Proof. Obviously, F has a least element 0F, since it is finite. Moreover, we can assume without loss of generality that F has a greatest element 1F: If not, there is an upper bound s for F within S by Corollary 4.7 as S is distributive. Clearly, Fs:= F ∪{s} is a subsemilattice of S extending F , and we can proceed by replacing F by Fs.

Suppose F is not distributive. Then the embedding OF: F → LF cannot be surjective by Corollary 4.8, so there exists a down-set H ⊆ J (F ) such that H /∈ imOF. Pick H0 minimal with this property. This means that H0 = ˆw for any w ∈ F , but H = wH for a unique wH whenever H is a down-set in J (F ) strictly contained in H0. Let {j1, . . . , jr} be the complete list of all maximal elements in H0. It follows that r ≥ 2 for otherwise H0 = j1. Put u = supFH0= supF{j1, . . . , jr}, which exists, since F has a greatest element. Note that u /∈ J (F ): u = jkfor all 1≤ k ≤ r, since r ≥ 2, so if u ∈ J (F ) with lower cover u−, then u−≥ jk for all 1≤ k ≤ r, contradicting u = supFH0.

Let U = { x ∈ F : ˆx ⊇ H0}, L = { x ∈ F : ˆx ⊆ H0}, and I = F \ (U ∪ L). Note that U is a nonempty up-set in F (1F ∈ U), L is nonempty down-set in F (0F ∈ L), and L ∩ U = ∅ by the choice of H0. Also, I = ∅, since otherwise H0 = ˆu. Pick x ∈ I. It follows that ˆx ∩ H0 ⊂ H0, and thus ˆx ∩ H0 =wx for some wx ∈ L. So wx = x and wx ⊆ ˆx, which implies wx < x, since ˆ is an embedding of F into LF by Corollary 4.8. Further, consider jk with 1 ≤ k ≤ r. Since jk ⊆ H0, we have ˆx ∩ jk ⊆ ˆx ∩ H0 = wx. Invoking the embedding property of ˆagain, we conclude that x ∧ jk≤ wx.

Using distributivity of S and Lemma 4.6(i), we find, for 1 ≤ k ≤ r, an element ak ∈ S satisfying ak ≥ jk and ak∧ x = wx< x. By Lemma 4.6(ii), we then find bx∈ S satisfying bx≥ ak (≥ jk) for 1≤ k ≤ r and bx∧ x = wx; moreover, bx x for otherwise wx= bx∧ x = x.

Define b ∈ S by b = x∈Ibx ∧ u. We claim that for any y ∈ F , either b ∧ y = b or b ∧ y ∈ F : If y ∈ U , then y ≥ u ≥ b, and thus b ∧ y = b. If y ∈ L, then also b ∧ y ∈ L, since L is a down-set in F . If y ∈ I, then

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b ∧ y ≤ by∧y = wy∈ L, and thus b∧y ∈ L again. It follows that Fb:= F ∪{b} is a subsemilattice of S containing F .

Consider j ∈ J (F ) such that j ≤ b: If j ∈ U , then j ≥ u, in fact, j > u, since u /∈ J (F ). Hence, j  b ≤ u. If j ∈ L, then j ≤ u and j ≤ jk for some 1 ≤ k ≤ r, which implies j ≤ bx for all x ∈ I, thus j ≤ b. If j ∈ I, then j  bj as shown above, thus j  b. Summing up, the down-set induced by b in J (F ) is L ∩ J (F ) = H0. Moreover, OF ∪ {(b, H0)} obviously is the canonical isomorphismOFb between Fband the subsemilattice generated within LF = O(J (F )) by im OF ∪ {H0}. Repeat the procedure with Fband iterate; the process breaks off with an isomorphism between some subsemilattice F0 of S and LF, making F0distributive.  Since our concern is with p-semilattices, the natural question is whether any finite p-subsemilattice F of a distributive p-semilattice P can be extended, within P , to a finite distributive p-subsemilattice F of P .

The starting observation is that LF = O(J (F )), as a finite distributive lattice, is pseudocomplemented for any meet-semilattice F . Indeed, for any down-set H ⊆ J (F ) the set H+:={ j ∈ J (F ) : ˆj∩ H = ∅ } is its pseudocom-plement.

Lemma 4.10. Let F be a finite p-semilattice. Then the canonical embedding OF: F → LF preserves pseudocomplements.

Proof. We have to show that x= ˆx+

for all x ∈ F . Now, j ≤ x∗iff j ∧ x = 0 for all j ∈ J (F ), and thus

ˆ

x+={ j ∈ J (F ) : ˆj∩ ˆx = ∅ } = { j ∈ J (F ) : j ∧ x = 0 }

={ j ∈ J (F ) : j ≤ x∗} = x. 

This is enough to prove the following result.

Corollary 4.11. Assume P is a distributive p-semilattice and F a finite p-subsemilattice of P . Then there exists a finite distributive p-semilattice F0 such that F  F0 P . In fact, we find such F0 satisfying F0=LF.

Proof. Consider Fb as in the proof of Proposition4.9 (note that 1F exists and equals 1P). All we need to show is that Fb is closed under pseudocomplements and thatOF∪{(b, H0)} preserves pseudocomplements. We claim that b∗= u∗, and thus b∗ ∈ F ⊆ Fb. Indeed, since b ≤ u, we have (1): b∗ ≥ u∗. Further, jk ≤ b (1 ≤ k ≤ r), thus jk∗ ≥ b∗, and so (2): j1∗∧ · · · ∧ jr ≥ b∗. Also, j1∗∧ · · · ∧ jr∗ ≤ jk (1 ≤ k ≤ r), hence (j1∗∧ · · · ∧ j∗r) ≥ jk∗∗ ≥ jk for all k, which implies (j1∗∧ · · · ∧ jr∗)∗≥ u, and finally (3): (j1∗∧ · · · ∧ jr∗)∗∗≤ u∗. But certainly, (4): j1∗∧ · · · ∧ jr∗≤ (j1∗∧ · · · ∧ jr∗)∗∗. Putting all together, we obtain

j1∗∧ · · · ∧ jr∗≤ (j∗1∧ · · · ∧ jr∗)∗∗≤ u∗≤ b∗≤ j1∗∧ · · · ∧ jr∗,

using (4), (3), (1), and (2), respectively; this proves our claim. Finally, that OF∪ {(b, H0)} preserves pseudocomplements is immediate. 

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5. Making the dense filter boolean

Suppose F0 is a finite distributive subsemilattice of a distributive p-semilattice P . The purpose of this section is to show that we can find, provided P satisfies a certain condition (A1), a finite distributive p-subsemilattice F1P such that F0 F1 P and the dense filter D(F1) is boolean.

We start by characterizing finite distributive p-semilattices F with boolean dense filters D(F ) in terms of their associated posets J (F ). Write J (F )min for the set of all minimal elements ofJ (F ).

Lemma 5.1. The dense filter of a finite distributive p-semilattice F is boolean iffJ (F ) \ J (F )min is an antichain.

Proof. Using F ∼=LF = O(J (F )), it is immediate that J (F )min represents the unique minimal dense element of F , and that D(F ) is isomorphic to the collection of all down-sets H ⊆ J (F ) containing J (F )min(under set inclusion), which in turn is isomorphic to the collection of all down-sets inJ (F )\J (F )min (again under set inclusion). But for any finite poset Q, one has that O(Q) is

boolean iff Q is an antichain. 

Consider any poset Q with order relation ≤ and a, b ∈ Q such that b covers a w.r.t. to ≤. It is easy to see that ≤:=≤ \{(a, b)} is also an order relation on Q: Dropping (a, b) from ≤ does neither affect reflexivity nor antisymmetry, and since b covers a w.r.t. ≤, (a, b) cannot be forced back into ≤ by applying transitivity to≤. We will use the short-hand notation Qab for the resulting poset (Q; ≤). We want to describeO(Qab):

Lemma 5.2. Consider Q, a, b as above and let M the uniquely determined maximal down-set inO(Q) not containing a. Then

O(Q

ab) =O(Q) ∪ { U ∪ {b} : U ∈ O(Q) and ↓b ∩ M ⊆ U ⊆ M } . Proof. SinceO(Qab) ⊇ O(Q) is clear, a down-set V ∈ O(Qab)\ O(Q) must contain b but not a. If x ∈ V \{b}, then x ∈ M , for otherwise, x ≥ a, putting a in V . Moreover, b >x implies b > x, and thus x ∈ V , so V \{b} ⊇ ↓b∩M . 

The property (A1) of p-semilattices is defined as follows: (∀d0, d1, d2∈ D(P ), t ∈ P )(∃x ∈ P )  (d0< d1< d2& t ∧ d0< t ∧ d1< t ∧ d2) = (A1) (d0< x < d2& x ∧ d1= d0& t ∧ d0< t ∧ x < t ∧ d2) . Our present aim is to prove

Lemma 5.3. Let P be a distributive p-semilattice satisfying (A1) and F  P a finite distributive p-subsemilattice of P . Let there be j1, j2∈ J (F )\J (F )min such that j2 covers j1. Then there exists a finite distributive p-semilattice F satisfying F  F P such that J (F) is order-isomorphic toJ (F )j

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Proof. Let H0 be the unique maximal down-set in J (F ) not containing j1. Certainly, H0⊇ J (F )min, so H0 corresponds to an element d0 ∈ D(F ) under the canonical isomorphism OF: F → LF, that is, d0 = H0. It follows that w  j1 iff w ≤ d0, for any w ∈ F . Note that if j ∈ J (F ) and j < j2, then either j = j1 or j ≤ d0 (since j2 covers j1 in J (F )). Hence, there are d1, d2∈ D(F ) such that d1= H0∪ {j1} and d2= H0∪ {j1, j2}.

We conclude that (i) d0< d1< d2and (ii) j2∧ d0< j2∧ d1< j2∧ d2= j2. Using (A1) with t = j2, we thus find x ∈ P such that d0< x < d2, d1∧x = d0, and j2∧ d0< j2∧ x < j2∧ d2. Note that x ∈ D(P ), since x > d0∈ D(F ) ⊆ D(P ).

Let F =F ∪ {x}P be the p-subsemilattice of P generated by F and x. Now, F ∪ { w ∧ x : w ∈ F } ⊆ P is obviously closed under meets; moreover, (w ∧ x)∗ = (w ∧ x)∗∗∗ = (w∗∗ ∧ x∗∗) = (w∗∗ ∧ 1)∗ = w∗∗∗ = w ∈ F as x ∈ D(P ). We conclude that F= F ∪ { w ∧ x : w ∈ F }.

We analyze the structure of F\ F : Suppose w ∈ F but w ∧ x /∈ F . Since x ≤ d2, we have w ∧ x = w ∧ d2∧ x, hence—replacing w by w ∧ d2—we can assume without loss of generality that w ≤ d2. So let w ≤ d2 and assume, towards a contradiction, that w ≤ d1. Then w ∧ x ≤ d1∧ x = d0, and thus w ∧ x = w ∧ x ∧ d0= w ∧ d0∈ F , contradicting w ∧ x /∈ F . So we can assume without loss of generality that w  d1. But, working in LF0, ˆw ⊆ d2 and

ˆ

w  d1 are equivalent to j2∈ ˆw, which translates into j2≤ w. Summing up, F\ F ⊆ { w ∧ x : j2≤ w ≤ d2}.

Conversely, let w ∈ F , j2 ≤ w ≤ d2, and suppose w ∧ x ∈ F . Then also j2∧ w ∧ x = j2∧ x ∈ F (as j2 ≤ w). Now, j2∧ d0 < j2∧ x < j2∧ d2 = j2 by (A1). Looking at LF, it is immediate that the unique down-set in J (F ) situated strictly betweenj2∩ d0andj2is given by (j2∩ d0)∪ j1, and thus contains j1. Translated back to F , this means that j1 ≤ j2∧ x, and thus j1≤ x. But also j1 ≤ d1 by construction of d1, so j1 ≤ x ∧ d1 = d0, contradicting the choice of d0. We conclude that w ∧ x /∈ F whenever w ∈ F and j2≤ w ≤ d2. Summing up, we have established F\ F = { w ∧ x : j2≤ w ≤ d2}.

Finally, assume w1, w2∈ F , j2 ≤ w1, w2 ≤ d2, and w1∧ x = w2∧ x. This implies w1∧ d0 = w1∧ x ∧ d1 = w2∧ x ∧ d1= w2∧ d0. Working in LF, we have d2\ d0={j1, j2}, and we conclude, observing that j1≤ j2≤ w1, w2, that 

w1= (w1∩ d0)∪ {j1, j2} = (w2∩ d0)∪ {j1, j2} = w2, that is, w1= w2. Define a map h from the interval [j2, d2]⊆ F to F\ F by h(w) = w ∧ x. So h is onto and injective by the above, and clearly order-preserving. Assume w ∧ x ≤ w∧ x. Then w ∧ w∧ x = w ∧ x, and thus w ∧ w= w or w ≤ w. So h−1 also preserves order and the final result is that h is an order-isomorphism between [j2, d2]⊆ F and F\ F .

Using h, we define a map φ : F → BF =P(J (F )) by φ(z) =  ˆ z for z ∈ F, ˆ w \ {j1} for z = h(w) ∈ F\ F.

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We want to determine φ[F]⊆ BF. Since φ(z) = OF(z) for z ∈ F , we certainly have LF ⊆ φ[F]. So consider z = h(w) ∈ F \ F . Then w ∈ [j2, d2] ⊆ F , and thus ˆw = ˆu ∪ {j1, j2} for a uniquely determined u ∈ [j2∧ d0, d0] ⊆ F , namely u = w ∧ d0. Consequently, φ(z) = ˆu ∪ {j2} ∈ BF \ LF. Conversely, any set ˆu ∪ {j2} ∈ BF with u ∈ [j2∧ d0, d0] has a unique preimage under φ, given by z = w ∧ x where ˆw = ˆu ∪ {j1, j2}. Since h is an order isomorphism, φ restricted to F\ F thus provides an order isomorphism between F\ F and { ˆu ∪ {j2} : u ∈ [j2∧ d0, d0]} (the latter ordered by set inclusion).

Putting this all together, we see—where ∼= is an order isomorphism— F∼=LF∪ { U ∪ {j2} : U ∈ O(J (F )) and ↓j2∩ H0⊆ U ⊆ H0} . By Lemma 5.2, the latter is justO((J (F )j

1j2), a (distributive) down-set lat-tice. An order isomorphism between lattices is always a lattice isomorphism, so F∼=O(J (F )j1j2) as lattices andJ (F) ∼=J (F )j1j2 as posets.  Corollary 5.4. Assume F0 is a finite distributive p-subsemilattice of a dis-tributive p-semilattice P satisfying (A1). Then there exists a finite disdis-tributive p-semilattice F1 such that F0 F1 P and the dense filter D(F1) is boolean. Proof. Let G0 = F0 and for i ≥ 0, obtain Gi+1 from Gi by applying Lemma 5.3 w.r.t. a covering pair of nonmimimal dense elements in Gi. The process stops when no such pair can be found; the final Gi0 has a boolean dense filter

by Lemma 5.1. Put F1= Gi0. 

6. Adding “central” elements to the skeleton

In this section, we assume that P is an arbitrary distributive p-semilattice, and F  P a finite distributive non-boolean p-subsemilattice whose dense filter D(F ) is boolean, that is, D(F ) ∼= 2n for some n ≥ 1. It follows that D(F ) contains n different maximally dense elements. Let Dmax(F ) = {d1, . . . , dn}.

Our purpose is to show that there exists a finite distributive p-subsemi-lattice FP such that F FP and F∼= 2ni=1Bˆifor some r ∈ ω and Bia boolean algebra for 1≤ i ≤ n, provided P satisfies the following property (A2) of p-semilattices:

(∀a ∈ Sk(P ), d, d∈ D(P ), p, p, x ∈ P )(∃z ∈ Sk(P )) 

(dd & p ≤ d& p ≤ d & p ≤ d& a ≤ d & a∗∧ p ≤ d & x∗≤ d) (A2) =⇒ (a ≤ z ≤ d & z∗∧ p ≤ d & z ∧ p ≤ d & (z ∧ x)∗≤ d) . Roughly speaking, boolean elements as provided by (A2) will be used to manufacture a finite extension Fof F containing a decomposition {u1, . . . , ut} of 1F = 1F into finitely many pairwise disjoint boolean elements such that

F/θui is isomorphic to either 2 or ˆBi for 1≤ i ≤ t.

We start by constructing, for every di ∈ Dmax(F ), an element ki ∈ Sk(P ) satisfying certain properties, using (A2). This is accomplished in several steps.

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Observe first that the set Hi:={ x ∈ F : x ≤ di} is closed under meets for every di ∈ Dmax(F ). Indeed, assume towards a contradiction that x, x ∈ Hi but x ∧ x ≤ di. By distributivity, we find y, y ∈ F such that y ≥ x, y≥ x, and y ∧ y= di. So, y, y ∈ D(F ), and by maximality of di, we have y = di or y = di. Without loss of generality, assume y = di. But this violates x ≤ di, proving our claim.

For 1≤ i ≤ n, define mi=Hi; it follows that mi is the smallest element of F not below di. Observe further that mi ≤ dj for any j = i, 1 ≤ j ≤ n: we have dj ≤ di by the maximality of dj, hence mi≤ dj by the minimality of mi. These properties of the di and mi together with (A2) prove the following lemma.

Lemma 6.1. Assume (A2) and suppose that k ∈ Sk(P ) and di ∈ Dmax(F ) satisfy k ≤ di and k∗∧ mi ≤ di. Then for any dj ∈ Dmax(F ) with j = i, there exists z ∈ Sk(P ) (depending on j) such that k ≤ z ≤ di, z∗∧ mi ≤ di, z ∧ mj ≤ dj, and satisfying that for all x ∈ F , x∗≤ dj⇒ (z ∧ x)∗≤ dj. Proof. Let Xj ={ x ∈ F : x∗≤ dj} = {xj1, . . . , xjn(j)}. Use (A2) with a := k, d := di, d := dj, p := mi, p:= mj, and x := xj1. (It is routine to check that the assumptions in (A2) are all satisfied). So, by (A2), there exists z1∈ Sk(P ) such that k ≤ z1≤ di, z1∗∧ mi ≤ di, z1∧ mj ≤ dj, and (z1∧ xj1)∗≤ dj.

Now apply (A2) with d, d, p, p as above but with a := z1 and x := xj2 (again, all the assumptions in (A2) are satisfied). So we find z2∈ Sk(P ) such that z1≤ z2≤ di, z2∗∧ mi ≤ di, z2∧ mj ≤ dj, and (z2∧ xj2)∗≤ dj. Note that z1 ≤ z2 implies z1∧ xj1≤ z2∧ xj1, and thus (z2∧ xj1) ≤ (z1∧ xj1) ≤ dj; consequently, z2 also satisfies (z2∧ xj1)∗≤ dj.

Continue until Xj is exhausted. The final zn(j) has all of the properties

required by the lemma, so put z = zn(j). 

Lemma 6.2. Assume (A2). Then for every di ∈ Dmax, there exists ki Sk(P ) such that ki ≤ di, ki∗∧ mi ≤ di, ki ∧ mj ≤ dj (for all j = i), and satisfying that for all x ∈ F , x∗≤ dj ⇒ (ki∧ x)∗≤ dj (for all j = i).

Proof. Assume, without loss of generality, that i = 1, and put h1 = 0. This means that h1 ∈ Sk(P ), h1 ≤ d1 and h∗1∧ m1 ≤ d1. Put j = 2 and apply Lemma 6.1 in order to obtain an element z satisfying 0 = h1 ≤ z ≤ d1, z∗∧ m1 ≤ d1, z ∧ m2 ≤ d2 and ∀x ∈ F (x∗ ≤ d2 ⇒ (z ∧ x)∗ ≤ d2). We put h2:= z.

Put j = 3 and repeat to obtain h3 ∈ Sk(P ) satisfying h2 ≤ h3 ≤ d1, h∗3∧ m1 ≤ d1, h3∧ m3 ≤ d3, and for all x ∈ F , x∗≤ d3⇒ (h3∧ x)∗≤ d3.

Now h3works also for d2: indeed, since h2≤ h3, we have h2∧m2≤ h3∧m2, and so h3∧ m2 ≤ d2, since h2∧ m2 ≤ d2. Similarly, for any x∗≤ d2, we have h2∧ x ≤ h3∧ x, and so (h3∧ x)∗≤ (h2∧ x)∗≤ d2.

Continuing, we finally obtain hn ∈ Sk(P ) satisfying hn ≤ d1, h∗n∧ m1 ≤ d1, hn∧ mj ≤ dj (for all j = 1), and for all x ∈ F , x∗≤ dj⇒ (hn∧ x)∗≤ dj) (for

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Note that k∗i ∧ mi ≤ di is equivalent to ki∗∧ z ≤ di for all z ∈ Hi. One direction is clear, since mi ∈ F and mi ≤ di. For the other, assume z ∈ F and z ≤ di. Then z ≥ mi by definition of mi, and thus ki∗∧ z ≥ k∗i ∧ mi. So if k∗i ∧ mi ≤ di, then k∗i ∧ z ≤ di a fortiori. The same argument shows that ki∧ mj ≤ dj (j = i) is equivalent to ki∧ z ≤ dj for all z ∈ Hj.

This gives the final description of the elements ki∈ Sk(P ) we are after. Lemma 6.3. For each element di ∈ Dmax(F ), there exists ki ∈ Sk(P ) such that

(i) ki≤ di,

(ii) for z ∈ F , z ≤ di ⇒ ki∗∧ z ≤ di,

(iii) for j = i and z ∈ F , z ≤ dj ⇒ ki∧ z ≤ dj, (iv) for j = i and x ∈ F , x∗≤ dj ⇒ (ki∧ x)∗≤ dj.

For easier reference, we list some consequences of the preceding lemma. Corollary 6.4. The elements ki described in Lemma 6.3 have the following additional properties:

(ii-bis) for z ∈ F , k∗i ∧ z ≤ di ⇒ z ≤ di,

(iii-bis) for j = i and z ∈ F , ki∧ z ≤ dj ⇒ z ≤ dj, (v) for y ∈ Sk(F ), y ≤ di ⇒ ki y ≤ di, (vi) for y ∈ Sk(F ), y ≤ dj ⇒ ki∗ y ≤ dj.

Proof. (ii-bis) and (iii-bis) are just the contrapositions of (ii) and (iii), respec-tively, in the preceding lemma.

(v): Assume y ∈ Sk(F ), y ≤ di, and put z = ki  y. Then k∗i ∧ z = ki∗∧ (ki y) = k∗i ∧ y ≤ y ≤ di. Using (ii-bis), we obtain z ≤ di as desired.

(vi): Assume y ∈ Sk(F ), y ≤ dj, and put z = k∗i  y. Then ki∧ z = ki∧ (ki∗ y) = ki∧ y ≤ y ≤ dj, which implies z ≤ dj, using (iii-bis).  Next, consider F [ki], the p-semilattice generated in P by F ∪ {ki}. Write Sk(F )[ki] for the (boolean) subalgebra of Sk(P ) generated by Sk(F ) ∪ {ki}; it is easy to see that Sk(F [ki]) = Sk(F )[ki]. Moreover, D(F [ki]) = D(F ). Since F is distributive, and thus representable by Lemma 4.4, it follows that also F [ki] is representable.

Lemma 6.5. F [ki] is distributive.

Proof. Using Lemma 4.5, it suffices to show that each d ∈ D(F [ki]) = D(F ) is distributive. But D(F ) is boolean and finite, so every d ∈ D(F ) is the meet of all dj ∈ Dmax(F ) covering d. Since the meet of distributive elements is always distributive, it remains to check that every dj∈ Dmax(F ) is distributive in F [ki].

We have F [ki] ={ b ∧ d : b ∈ Sk(F [ki]), d ∈ D(F [ki])}, since F [ki] is repre-sentable. Using conjunctive normal form for boolean terms and D(F [ki]) = D(F ), this boils down to

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So assume dj ≥ v ∧ w with dj ∈ Dmax(F ) and v, w ∈ F [ki]. We want to find v, w ∈ F [ki] such that v ≥ v, w ≥ w, and v∧ w = dj. Explicitly, v, respectively, w is given as v = (v1 ki)∧ (v2 ki)∧ dv, respectively, w = (w1 ki)∧ (w2 k∗i)∧ dw with v1, v2, w1, w2∈ Sk(F ) and dv, dw∈ D(F ).

Case 1: j = i. We have

dj ≥ ((v1 ki)∧ (v2 k∗i)∧ dv)∧ ((w1 ki)∧ (w2 ki)∧ dw)

= ((v1∧ w1) ki)∧ ((v2∧ w2) ki)∧ dv∧ dw≥ ki∧ (v2∧ w2)∧ dv∧ dw. Putting z = (v2∧w2)∧dv∧dw, we have dj≥ ki∧z. Observe that z ∈ F ; so Corollary 6.4(iii-bis) applies and gives z ≤ dj, that is, dj ≥ (v2∧w2)∧(dv∧dw). Using distributivity of F , we find d1 ≥ v2 ∧ w2, d2 ≥ dv ∧ dw such that d1∧ d2 = dj. This makes d1, d2 dense and thus d1 = dj or d2 = dj, since dj ∈ Dmax(F ).

Suppose dj = d2. Then dj ≥ dv∧ dw, and by distributivity of F again, we find dv ≥ dv, dw ≥ dw such that dv∧ dw = dj. By maximality of dj, we must have dv = dj or dw= dj. In the first case, we obtain dj = dv≥ dv ≥ v. Putting v = dj, w = 1, we realize v ≥ v, w ≥ w, and v ∧ w = dj, as desired. If dw = dj, the analogous argument shows that v = 1 and w = dj work as well.

It remains to consider the case dj = d1. This time, we have dj ≥ v2∧ w2, and distributivity of F provides dv ≥ v2, dw ≥ w2 such that dv∧ dw = dj. Again, dj= dv, and thus dj≥ v2, or dj = dw, and thus dj≥ w2.

Use Lemma 6.3(iv) with x = 1 to obtain dj ≥ k∗i. Applying Corol-lary 6.4(vi), we deduce that dj ≥ v2 k∗i or dj ≥ w2 k∗i. In the first case, dj ≥ (v2ki)∧(v1ki)∧dv = v; in the second, dj ≥ (w2k∗i)∧(w1ki)∧dw= v. This shows that v= djand w = 1, respectively v= 1 and w= dj, have the desired properties.

Case 2: j = i. The arguments have the same structure as in the case j = i, so we give only an outline. Start from

di≥ ((v1 ki)∧ (v2 ki)∧ dv)∧ ((w1 ki)∧ (w2 k∗i)∧ dw)

= ((v1∧ w1) ki)∧ ((v2∧ w2) k∗i)∧ dv∧ dw≥ (v1∧ w1)∧ ki∗∧ dv∧ dw. Put z = (v1∧ w1)∧ dv∧ dw to obtain di ≥ ki∗∧ z. Since z ∈ F , Corollary 6.4(ii-bis) applies and gives z ≤ di, that is, di ≥ (v1∧ w1)∧ (dv ∧ dw). By distributivity of F , find d1 ≥ v1∧ w1, d2 ≥ dv∧ dw such that d1∧ d2 = di, thus d1= di or d2= di.

If di = d1, obtain v = di and w = 1, respectively v = 1 and w = di, as in Case 1. If di = d2, the same arguments work, using Lemma 6.3(i) and

Corollary 6.4(v). 

We are now ready to construct F P such that we have F  F P and F = 2ni=1Bˆi. Observe that F [ki] is a finite distributive p-subsemilattice of P containing F and having the same dense filter as F . So we can iterate the construction of F [ki] with some other di ∈ Dmax(F ), finding an element

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ki∈ Sk(P ) that has all the required properties with respect to F [ki] and thus,

a fortiori, with respect to F .

Explicitly, let G0 = F and for 1 ≤ i ≤ n, put Gi = Gi−1[ki]. Then define F = Gn. F is distributive, D(F) = D(F ), and in particular, F has the following properties: for every element di ∈ Dmax(F), there is ki ∈ Sk(F) with

(i) ki≤ di by 6.3(i),

(ii) ki∗ ≤ di by 6.3(ii), setting z = 1,

(iii) ki ≤ dj by 6.3(iii) for j = i, setting z = 1, (iv) ki∗≤ dj by 6.3(iv) for j = i, setting x = 1.

Proposition 6.6. For some r ∈ ω, F = 2ni=1Bˆi, with Bi a boolean algebra for 1≤ i ≤ n.

Proof. Let C = { a ∈ Sk(F) : a ≤ d ∈ D(F) implies d = 1 }. C = ∅, since 1∈ C. Moreover, C is closed under meets. Let a, b ∈ C and a∧b ≤ d ∈ D(F). By distributivity, we find da, db such that da ≥ a, db ≥ b, and da∧ db = d. Hence, da, db ∈ D(F),and so da = db = 1, thus d = 1. Since F is finite, c0:=



C exists and is the smallest element of C. Note that c0 = 0 (otherwise F would be boolean).

The elements of F have a canonical form. For b ∈ Sk(F), let Δb = { dl∈ Dmax: b ≤ dl}. Since F is representable and D(F) is boolean, it is clear that for every u ∈ F, there exists a representation u = b ∧Q with b ∈ Sk(F) and some subset Q ⊆ Δb (note thatQ = 1 iff Q = ∅). Assume u = b∧Q = b∧Qare two different representations of this type. Applying ∗∗, we obtain u∗∗= b = b, so Q = Q, and we find, without loss of generality, an element d ∈ Q \ Q. Note that Q = ∅ for otherwise Q = 1, and so b = b ∧ 1 = b ∧Q, implying b ≤ Q ≤ d. Writing Q = {d1, . . . , dt}, we obtain d ≥ b ∧ d1∧ · · · ∧ dt. By distributivity, there are v, w ∈ F such that v ≥ b, w ≥ d1 ∧ · · · ∧ dt, and v ∧ w = d. By maximality of d, it follows that v = d or w = d. But v = d is not possible, since d ≥ b, so we must have w = d, that is, d ≥ d1∧ · · · ∧ dt. This implies the existence of y, z such that y ≥ d1, z ≥ d2∧ · · · ∧ dt, and y ∧ z = d. Repeat the procedure, using distributivity successively, to obtain finally that d = ds for some ds ∈ Q, which contradicts our choice of d. It follows that Q = Q. So there is a unique subset Qu⊆ Δu∗∗ such that u = u∗∗∧Qu, and obviously

Qu = { dl∈ Dmax: u ≤ dl and u∗∗ ≤ dl}. Consequently, the correspondence u ←→ (u∗∗, Qu) is bijective.

For 1≤ i ≤ n, define ai = ki c∗0 ∈ Sk(F). We claim that ai aj = 1 for i = j: Let ai aj ≤ d ∈ D(F). If d = 1, there exists dk ∈ Dmax(F) such that d ≤ dk, implying ki ≤ ai ≤ dk and kj ≤ aj ≤ dk. By (iii) above, we have i = k and j = k, contradicting i = j. Thus, d = 1 and, consequently, ai aj≥ c0by the definition of c0. But ai, aj≥ c∗0 by the definition of the ai, so ai aj ≥ c0 c∗0= 1, as claimed.

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By definition, a∗i = (ki c∗0) = k∗i ∧ c0. Note that a∗i = 0: Otherwise, ki∗∧ c0 = 0, which implies c0 ≤ k∗∗i = ki ≤ di (by (i) above), contradicting the definition of c0. We have a∗i ∧ a∗j = (a∗i ∧ a∗j)∗∗= (ai aj) = 1= 0 for i = j. Moreover, a∗i ∧ c∗0= k∗i ∧ c0∧ c∗0= 0 for 1≤ i ≤ n.

On the other hand, a∗1 · · ·  a∗n = (k∗1 · · ·  kn∗)∧ c0 ≤ c0. If we have that (k∗1 · · ·  kn)∧ c0 < c0, there exists 1 = d ∈ D(F)—and with that, dl∈ Dmax(F)—such that (k∗1· · ·kn∗)∧c0≤ d ≤ dl. Using distributivity, we find d1≥ k1∗· · ·k∗nand d2≥ c0such that d1∧d2= dl. Hence, d1, d2∈ D(F), and thus d2= 1, which gives d1= dl. But dl≥ k1∗ · · ·  kn∗ implies dl≥ k∗l, contradicting (ii) above. Consequently, a1· · ·a∗n = (k1∗· · ·k∗n)∧c0= c0. Summing up, we see that{c∗0, a1, . . . , an} provides a boolean partition of 1.

We next determine the structure of the factor algebras F/θa∗

i for 1≤ i ≤ n.

Now, F/θa∗

i = { u ∧ a∗i : u ∈ F} (the latter with the operations given in

Section 2). So we have to compute the meets u∗∗∧Qu∧ a∗i for u ∈ F. Now, a∗i = k∗i ∧ c0, and k∗i ≤ dl for l = i by (iv) above, so u ∧ a∗i = u∗∗∧ a∗i if di ∈ Qu, and u ∧ a∗i = u∗∗∧ a∗i ∧ di if di ∈ Qu. We distinguish the cases u∗∗≥ a∗i, respectively, u∗∗ ≥ a∗i.

First, assume that u∗∗≥ a∗i. Then u∧a∗i = a∗i if di ∈ Qu, and u∧a∗i = a∗i∧di if di ∈ Qu. We claim that a∗i ∧ di < ai∗. If not, a∗i = k∗i ∧ c0≤ di. But then, by distributivity, there are v, w such that k∗i ≤ v, c0 ≤ w, and v ∧ w = di, implying v = di or w = di. Now w = di yields c0 ≤ di, which is not possible, and v = dimeans k∗i ≤ di, violating (ii). Thus, a∗i∧ di< a∗i as claimed. Since (a∗i ∧ di)∗∗= ai∗, we see that a∗i ∧ di ∈ Sk(F).

Next, suppose u∗∗ ≥ a∗i. Then u∗∗ ∧ a∗i < a∗i. We have (u∗∗ ∧ a∗i) ki ≥ c0. (Since meeting (u∗∗∧ a∗i) ki ≥ c0 on both sides with k∗i gives u∗∗ ∧ a∗i ∧ ki∗ ≥ k∗i ∧ c0 = a∗i, violating u∗∗∧ a∗i < a∗i.) So there exists dl such that dl ≥ (u∗∗∧ a∗i) ki, whence dl ≥ ki, and thus l = i by (iii). We conclude that di≥ (u∗∗∧ a∗i) ki ≥ u∗∗∧ a∗i. So u∗∗∧ a∗i ≤ a∗i ∧ di (in fact, u∗∗∧a∗i < a∗i∧di, since a∗i∧di is non-boolean), and we obtain u∧a∗i = u∗∗∧a∗i whether di∈ Quor not.

In other words,{ u ∧ a∗i : u ∈ F} consists of a∗i, ai∧ di, and all b ∈ Sk(F) with b < a∗i, and the latter all satisfy b ≤ ai∗∧ di. Moreover, a∗i ∧ di is the only non-boolean element occurring. It follows that F/θa∗

i = ˆBi with Bi a

finite boolean algebra.

It remains to compute F/θc∗

0. Consider dl∈ Dmax(F). Clearly, dl≥ 0 = c0∧ c∗0, so there are u, v such that u ≥ c0, v ≥ c∗0, and u ∧ v = dl. This implies u = 1 (since u ∈ D(F) and c0∈ C), thus v = dl. This shows that dl≥ c∗0 for all dl∈ Dmax(F).

Again, F/θc∗

0 ={ u ∧ c∗0: u ∈ F} (the latter with the operations given in Section 2). So we have to compute the meets u∗∗∧Qu∧ c∗0 for u ∈ F. But since dl≥ c∗0 for any dl∈ Dmax(F), so u∗∗∧



Qu∧ c∗0= u∗∗∧ c∗0, and so we have u ∧ c∗0= u∗∗∧ c∗0 for all u ∈ F, whence

{ u ∧ c∗

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It follows that F/θc∗

0 is a finite boolean algebra. Note that F/θc∗0 is the trivial one-element algebra if c0= 1.

Let the canonical homomorphism h : F → F/θc∗

0×

n

i=1F/θa∗

i be given

by h(u) := (u ∧ c∗0, u ∧ a∗1, . . . , u ∧ a∗n); it is injective and surjective.

For injectivity, consider v, w ∈ F, v = w. Suppose first that v∗∗ = w∗∗. Since{c∗0, a1, . . . , an} is a partition, so v∗∗∧c∗0 = w∗∗∧c∗0or v∗∗∧a∗l = w∗∗∧a∗l for some 1≤ l ≤ n. In the first case, we are done, since u∧c∗0= u∗∗∧c∗0 for all u ∈ F, and thus v ∧ c∗0 = w ∧ c∗0. In the second, remember that u ∧ a∗i equals a∗i or a∗i∧ diif u∗∗≥ a∗i, and u∗∗∧ ai∗if u∗∗ ≥ a∗i (and then u∗∗∧ a∗i < a∗i∧ di). Since v∗∗∧ a∗l = w∗∗∧ al∗, we cannot have v∗∗, w∗∗≥ a∗l, so suppose, without loss of generality, that v∗∗ ≥ a∗l, which implies v ∧ a∗l = v∗∗ ∧ a∗l. If also w∗∗ ≥ a∗l, then w ∧ a∗l = w∗∗∧ a∗l = v∗∗∧ a∗l = v ∧ a∗l, and we are done. If w∗∗≥ a∗l, then v ∧ a∗l = v∗∗∧ a∗l < a∗l ∧ dl≤ w ∧ a∗l, settling also this case.

Now suppose v∗∗= w∗∗. This implies Qv = Qw, so assume, without loss of generality, that there is dl∈ Qw\Qv. It follows that v ≤ dlbut w ≤ dl.We infer that v ∧ a∗l ≤ a∗l ∧ dl. Suppose, towards a contradiction, that v ∧ a∗l = w ∧ a∗l. Then w ∧ a∗l ≤ dl and, by distributivity, we find x, y such that x ≥ v,y ≥ a∗l and x ∧ y = dl. As usual, we must have x = dl or y = dl, which forces the contradiction v ≤ dl, respectively, a∗l ≤ dl. Thus,v ∧ a∗l = w ∧ a∗l, as desired.

For surjectivity, consider, without loss of generality, w = (b0, b1, . . . , bk, bk+1∧ dk+1, . . . , bn∧ dn) in F/θc∗

0 ×

n

i=1F/θa∗i with b0, . . . , bn ∈ Sk(F), b0 ≤ c0, and bj ≤ a∗j for

1 ≤ j ≤ n, and dl ∈ Dmax(F) for k + 1 ≤ l ≤ n. It follows that w∗∗ = (b0, . . . , bn). Put x = (b0 · · ·  bn)∧ dk+1∧ · · · ∧ dn. Then h(x) = w.  Corollary 6.7. Assume P is an arbitrary distributive p-semilattice satisfying (A2), and F1 P a finite distributive p-semilattice such that D(F1) ∼= 2n for some n ≥ 1. Then there exists a finite distributive p-semilattice F2 such that F1 F2 P and F2= 2

n

i=1Bˆi, for some r ∈ ω and Bi a boolean algebra for 1≤ i ≤ n.

Proof. Use F2:= F as provided by Proposition 6.6. 

7. Extending factors ˆFn to ˆA

Assume that P is an arbitrary distributive p-semilattice, F  P a finite distributive p-semilattice of the form F ∼= 2si=1Bˆi for some r ∈ ω, and Bi a finite boolean algebra for 1 ≤ i ≤ s. We will show that F can be extended to a p-semilattice F such that F  F P and F∼= 2r× ( ˆA)s(with Athe countable atom-free boolean algebra), provided P satisfies the following property (A3): (∀b1∈ Sk(P ), d ∈ D(P ))(∃b2∈ Sk(P ))  b1< d < 1 =⇒ b1< b2< d & b1 b∗2 < d . (A3)

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The key ingredient needed to prove the above statement is contained in the following lemma.

Lemma 7.1. Let F ∼= ˆFk × F, with k ≥ 1 and F any finite distributive p-semilattice. If P is any distributive p-semilattice satisfying (A3) and F  P , there exists F+= ˆ

Fk+1× F such that F  F+ P . Such F+ can be obtained by “splitting” any atom of ˆFk.

Proof. Assume F ∼= ˆFk× F P and k ≥ 2. Let c = (1, 0) and c∗ = (0, 1) be the central elements of F associated with the direct product decomposition of F specified above. Pick an atom of F such that a ≤ c. It follows that a ∈ Sk(F ); moreover, a∗ is a coatom of Sk(F ) and a∗ ≥ c∗. Further, let e ∈ D(F ) be the unique dense element satisfying e = 1 and e ≥ c∗. Now use (A3) to find u∗∈ Sk(P ) such that a∗< u∗< e and a∗ u < e.

We have u < a, since u∗ > a∗, and u = 0 (for otherwise u∗ = 1 ≤ e), hence 0 < u < a. Consider a ∧ u∗: a ∧ u∗ = (a∗ u)∗ = 0 (for otherwise (a∗ u)∗∗ = a∗ u = 1 ≤ e), and a ∧ u∗ = u∗ (for otherwise u∗ ≤ a, whence a∗ ≤ u∗∗ = u < a, and thus a = 1). Summing up, we have 0 < a ∧ u∗ < u∗ and obviously u ∧ (a ∧ u∗) = 0 and u  (a ∧ u∗) = a. So, u and a ∧ u∗provide a proper splitting of the atom a.

Let F [u] be the p-semilattice generated F ∪ {u} within P . It is clear that F [u] is representable, being generated by Sk(F )[u] (the p-semilattice gener-ated by Sk(F ) ∪ {u} within Sk(P )) together with D(F ), and that Sk(F [u]) = Sk(F )[u]. So we start by describing Sk(F )[u].

Note that every x ∈ Sk(F ) has a unique representation x = x1 x2 with x1 ≤ c and x2 ≤ c∗: take x1 = x ∧ c and x2 = x ∧ c∗. The same holds for u with u1= u and u2= 0, and u∗with (u∗)1= u∗∧ c and (u∗)2= u∗∧ c∗= c∗. Define S ⊆ Sk(F )[u] by s ∈ S iff s = s1 s2, where s1 is x1 or x1 u or x1∧ u∗for some x1∈ F with x1≤ c, and where s2= x2for some x2∈ F with x2 ≤ c∗. It is routine to see that S is closed under ∧, , and∗ by checking cases (this boils down to checking that S1 ={ s1: s ∈ S } is closed under ∧, , and where s

1= s∗1∧ c). Moreover, S contains u, so S = Sk(F )[u]. For any member of D(F) ={ δt: t ∈ T }, define dt∈ D(F ) to be the dense element associated with (1, δt) in the direct product decomposition of F . It follows that every d ∈ D(F ) can be written as d1∧ d2 with d1 ∈ {e, 1} and d2= dtfor some t ∈ T . Finally, since F [u] is representable, any w ∈ F [u] can be written as w = s ∧ d1∧ d2 with s ∈ S and d1, d2 as specified just above.

We determine F [u]/θc ={ w ∧ c : w ∈ F [u] } (the latter under the opera-tions specified in Section 2). Now, w ∧ c = (s1 s2)∧ d1∧ d2∧ c, which reduces to s1∧ d1, since s1 ≤ c ≤ d2 and c ∧ s2 = 0. Let Q be the set of all atoms of Sk(F ) lying below c; then a ∈ Q and Q has k elements. If s1< c, then s1 is the boolean join of a proper subset of (Q \ {a}) ∪ {u, a ∧ u∗} that contains k + 1 elements, and we conclude that Sk(F [u]/θc) ∼= 2k+1. By construction, all such s1< c are below e ∧ c, which is thus the only non-boolean element in F [u]/θc. It follows that F [u]/θc∼= ˆFk+1.

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